Next, let \(\mathcal {M}(n)\)  denote the full subcategory of
\(\mathcal {O}_{n+1}\)  whose objects consist of the surjections
\(\phi \colon [n+1]\rightarrow [m]\)  with \(\phi (n) = \phi (n+1)\) . Note that \(s^n\colon [n+1]\rightarrow [n]\)  is an initial object of
\(\mathcal {M}(n)\) . We note the following fact about these subcategories
(compare with Lemma VII 1.26 of [1]}): the following diagram is a pushout in the category of small categories
\({\mathcal {O}_n [d] [r] & \mathcal {M}(n) [d] \\\sigma _0(\mathcal {O}_{n}) [r] & \mathcal {O}_{n+1} }\) 
